C0-estimates and Smoothness of Solutions to the Parabolic Equation Defined by Kimura Operators

Kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics. Their infinitesimal generator is an elliptic differential operator whose second-order coefficients matrix degenerates on the boundary of the domain. In this article, we consider the inhomogeneous initial-value problem defined by generators of Kimura diffusions, and we establish C-estimates, which allows us to prove that solutions to the inhomogeneous initialvalue problem are smooth up to the boundary of the domain where the operator degenerates, even when the initial data is only assumed to be continuous.